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vectors and rotations for three-dimensional space were rooted
hi his subconscious conviction that the most important laws of
common aigebra must persist in the algebra he was seeking.
How were vectors in three-dimensional space to be multiplied
To sense the difficulty of the problem it is essential to bear in
mind (see Chapter on Gauss) that ordinary complex numbers
a j. oi (z = V  1) had been given a simple interpretation in
terms of rotations in a plane, and further that complex numbers
obey all the rules of common algebra, in particular the commuta-
tive law of multiplication: it A, B are any complex numbers, then
A x B  B x A, whether A, B are interpreted algebraically,
or in terms of rotations in a plane. It was but human then to
anticipate that the same commutative law would hold for the
generalizations of complex numbers which represent rotations in
space of three dimensions.
Hamilton's great discoverjr - or invention - was an algebra,
one of the 'natural' algebras of rotations in space of three
dimensions, in which the commutative law of multiplication
does not hold. In this Hamiltonian algebra of quaternions (as
he called his invention), a multiplication appears in which
A ;< B is not equal to B x A but to minus B x A, that is,
A y. B = - B x A.
That a consistent, practically useful system of algebra could
be constructed in defiance of the commutative law of multipli-
cation was a discovery of the first order, comparable, perhaps,
to the conception of non-Euclidean geometry. Hamilton him-
self was so impressed by the magnitude of what suddenly
dawned on his mind (after fifteen years of fruitless thought) one
day (16 October 1843) when he was out walking with his wife
that he carved the fundamental formulae of the new algebra
in the .stone of the bridge on which he found himself at the
moment. His great invention showed algebraists the way to
other algebras until to-day, following Hamilton's lead, mathe-
maticians manufacture algebras practically at will by negating
one or more of the postulates.for a field and developing tht
consequences. Some of these "algebras'1 are extremely useful;
the general theories embracing swarms of them include Hamil-